Boosting the discriminatory power of sparse survival models via optimization of the concordance index and stability selection

Abstract

Background: When constructing new biomarker or gene signature scores for time-to-event outcomes, the underlying aims are to develop a discrimination model that helps to predict whether patients have a poor or good prognosis and to identify the most influential variables for this task. In practice, this is often done fitting Cox models. Those are, however, not necessarily optimal with respect to the resulting discriminatory power and are based on restrictive assumptions. We present a combined approach to automatically select and fit sparse discrimination models for potentially high-dimensional survival data based on boosting a smooth version of the concordance index (C-index). Due to this objective function, the resulting prediction models are optimal with respect to their ability to discriminate between patients with longer and shorter survival times. The gradient boosting algorithm is combined with the stability selection approach to enhance and control its variable selection properties.

Results: The resulting algorithm fits prediction models based on the rankings of the survival times and automatically selects only the most stable predictors. The performance of the approach, which works best for small numbers of informative predictors, is demonstrated in a large scale simulation study: C-index boosting in combination with stability selection is able to identify a small subset of informative predictors from a much larger set of non-informative ones while controlling the per-family error rate. In an application to discover biomarkers for breast cancer patients based on gene expression data, stability selection yielded sparser models and the resulting discriminatory power was higher than with lasso penalized Cox regression models.

Conclusion: The combination of stability selection and C-index boosting can be used to select small numbers of informative biomarkers and to derive new prediction rules that are optimal with respect to their discriminatory power. Stability selection controls the per-family error rate which makes the new approach also appealing from an inferential point of view, as it provides an alternative to classical hypothesis tests for single predictor effects. Due to the shrinkage and variable selection properties of statistical boosting algorithms, the latter tests are typically unfeasible for prediction models fitted by boosting.

Keywords: Boosting; Concordance index; High-dimensional data; Stability selection; Time-to-event data; Variable selection.

Fig. 1

Variable selection for the breast cancer application. Number of selected variables resulting from boosting a smooth version of the C-index (left boxplots) and Cox lasso (right boxplots) with and without stability selection for different values of π _thr. Boxplots refer to the results from 100 stratified subsamples drawn from the complete data set

Fig. 2

Discriminatory power for the breast cancer application. Resulting C-index on 100 test samples from the breast cancer application comparing both C-index boosting (left boxplots) and Cox lasso (right boxplots) with and without stability selection for different values of π _thr